Simplify The Expression: 2 X 8 X 3 + 5 12 X \frac{2x}{8x^3} + \frac{5}{12x} 8 X 3 2 X ​ + 12 X 5 ​

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Introduction

In mathematics, simplifying expressions is an essential skill that helps us solve problems efficiently. When dealing with fractions, combining them can be a bit challenging, but with the right approach, it becomes manageable. In this article, we will focus on simplifying the expression 2x8x3+512x\frac{2x}{8x^3} + \frac{5}{12x} using a step-by-step approach.

Understanding the Expression

The given expression consists of two fractions: 2x8x3\frac{2x}{8x^3} and 512x\frac{5}{12x}. To simplify this expression, we need to find a common denominator for both fractions. The common denominator is the least common multiple (LCM) of the denominators of both fractions.

Finding the Common Denominator

To find the LCM of 8x38x^3 and 12x12x, we need to factorize both numbers.

  • 8x3=23⋅x38x^3 = 2^3 \cdot x^3
  • 12x=22⋅3⋅x12x = 2^2 \cdot 3 \cdot x

The LCM of 8x38x^3 and 12x12x is 23⋅3⋅x3=24x32^3 \cdot 3 \cdot x^3 = 24x^3.

Simplifying the Expression

Now that we have the common denominator, we can rewrite both fractions with the common denominator.

  • 2x8x3=2x⋅38x3⋅3=6x24x3\frac{2x}{8x^3} = \frac{2x \cdot 3}{8x^3 \cdot 3} = \frac{6x}{24x^3}
  • 512x=5⋅2x12x⋅2x=10x24x2\frac{5}{12x} = \frac{5 \cdot 2x}{12x \cdot 2x} = \frac{10x}{24x^2}

Combining the Fractions

Now that both fractions have the same denominator, we can combine them by adding the numerators.

6x24x3+10x24x2=6x+10x24x2⋅x=16x24x3\frac{6x}{24x^3} + \frac{10x}{24x^2} = \frac{6x + 10x}{24x^2 \cdot x} = \frac{16x}{24x^3}

Simplifying the Result

We can simplify the result by dividing both the numerator and the denominator by their greatest common divisor (GCD).

The GCD of 16x16x and 24x324x^3 is 8x8x. Dividing both the numerator and the denominator by 8x8x, we get:

16x24x3=23x2\frac{16x}{24x^3} = \frac{2}{3x^2}

Conclusion

In this article, we simplified the expression 2x8x3+512x\frac{2x}{8x^3} + \frac{5}{12x} using a step-by-step approach. We found the common denominator, simplified the fractions, combined them, and finally simplified the result. This example demonstrates the importance of simplifying expressions in mathematics and how it can be achieved using the right techniques.

Frequently Asked Questions

Q: What is the common denominator of two fractions?

A: The common denominator is the least common multiple (LCM) of the denominators of both fractions.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you need to factorize both numbers and multiply the highest power of each prime factor.

Q: Can I simplify an expression with multiple fractions?

A: Yes, you can simplify an expression with multiple fractions by finding the common denominator, simplifying the fractions, combining them, and finally simplifying the result.

Tips and Tricks

  • When simplifying expressions, always look for common factors in the numerator and the denominator.
  • Use the least common multiple (LCM) to find the common denominator.
  • Simplify the fractions before combining them.
  • Use the greatest common divisor (GCD) to simplify the result.

Further Reading

References

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Introduction

In our previous article, we simplified the expression 2x8x3+512x\frac{2x}{8x^3} + \frac{5}{12x} using a step-by-step approach. However, we know that simplifying expressions can be a challenging task, especially when dealing with fractions. In this article, we will address some of the most frequently asked questions about simplifying expressions with fractions.

Q&A: Simplifying Expressions with Fractions

Q: What is the first step in simplifying an expression with fractions?

A: The first step in simplifying an expression with fractions is to find the common denominator. The common denominator is the least common multiple (LCM) of the denominators of both fractions.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you need to factorize both numbers and multiply the highest power of each prime factor.

Q: Can I simplify an expression with multiple fractions?

A: Yes, you can simplify an expression with multiple fractions by finding the common denominator, simplifying the fractions, combining them, and finally simplifying the result.

Q: What is the difference between simplifying and combining fractions?

A: Simplifying fractions involves reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). Combining fractions involves adding or subtracting fractions with different denominators by finding the common denominator.

Q: How do I simplify a fraction with a variable in the denominator?

A: To simplify a fraction with a variable in the denominator, you need to factorize the denominator and cancel out any common factors between the numerator and the denominator.

Q: Can I simplify an expression with negative fractions?

A: Yes, you can simplify an expression with negative fractions by following the same steps as simplifying positive fractions.

Q: How do I simplify an expression with fractions and variables in the numerator and denominator?

A: To simplify an expression with fractions and variables in the numerator and denominator, you need to factorize the numerator and the denominator and cancel out any common factors.

Tips and Tricks

  • When simplifying expressions, always look for common factors in the numerator and the denominator.
  • Use the least common multiple (LCM) to find the common denominator.
  • Simplify the fractions before combining them.
  • Use the greatest common divisor (GCD) to simplify the result.
  • Factorize the numerator and the denominator to cancel out any common factors.

Common Mistakes to Avoid

  • Not finding the common denominator before combining fractions.
  • Not simplifying the fractions before combining them.
  • Not using the greatest common divisor (GCD) to simplify the result.
  • Not factorizing the numerator and the denominator to cancel out any common factors.

Further Reading

References

Conclusion

Simplifying expressions with fractions can be a challenging task, but with the right approach and techniques, it becomes manageable. By following the steps outlined in this article, you can simplify expressions with fractions and variables in the numerator and denominator. Remember to always look for common factors, use the least common multiple (LCM) to find the common denominator, and simplify the fractions before combining them.